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Stationarity and Regularity of Infinite Collections of Sets

Author

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  • Alexander Y. Kruger

    (University of Ballarat)

  • Marco A. López

    (University of Ballarat
    University of Alicante)

Abstract

This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces.

Suggested Citation

  • Alexander Y. Kruger & Marco A. López, 2012. "Stationarity and Regularity of Infinite Collections of Sets," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 339-369, August.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:2:d:10.1007_s10957-012-0043-4
    DOI: 10.1007/s10957-012-0043-4
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    References listed on IDEAS

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    1. Jonathan M. Borwein & Alejandro Jofré, 1998. "A nonconvex separation property in Banach spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 48(2), pages 169-179, November.
    2. A. Moldovan & L. Pellegrini, 2009. "On Regularity for Constrained Extremum Problems. Part 1: Sufficient Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 147-163, July.
    3. A. Moldovan & L. Pellegrini, 2009. "On Regularity for Constrained Extremum Problems. Part 2: Necessary Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 165-183, July.
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    Cited by:

    1. Alexander Y. Kruger & Nguyen H. Thao, 2015. "Quantitative Characterizations of Regularity Properties of Collections of Sets," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 41-67, January.
    2. Hoa T. Bui & Alexander Y. Kruger, 2019. "Extremality, Stationarity and Generalized Separation of Collections of Sets," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 211-264, July.
    3. Alexander Y. Kruger & Marco A. López, 2012. "Stationarity and Regularity of Infinite Collections of Sets. Applications to Infinitely Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 390-416, November.

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